Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{-2}z^{-5})^{2}}}{{(a^{2}z^{-3})^{3}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{-2}z^{-5})^{2} = (a^{-2})^{2}(z^{-5})^{2}}$ On the left, we have ${a^{-2}}$ to the exponent ${2}$ . Now ${-2 \times 2 = -4}$ , so ${(a^{-2})^{2} = a^{-4}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{-2}z^{-5})^{2}}}{{(a^{2}z^{-3})^{3}}} = \dfrac{{a^{-4}z^{-10}}}{{a^{6}z^{-9}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-4}z^{-10}}}{{a^{6}z^{-9}}} = \dfrac{{a^{-4}}}{{a^{6}}} \cdot \dfrac{{z^{-10}}}{{z^{-9}}} = a^{{-4} - {6}} \cdot z^{{-10} - {(-9)}} = a^{-10}z^{-1}$